Geometric and homological properties of affine Deligne-Lusztig varieties

Abstract

This paper studies affine Deligne-Lusztig varieties Xw(b) in the affine ag variety of a quasi-split tamely ramified group. We describe the geometric structure of Xw(b) for a minimal length element w in the conjugacy class of an extended affine Weyl group. We then provide a reduction method that relates the structure of Xw(b) for arbitrary elements w in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of Görtz, Haines, Kottwitz and Reuman. © 2014 Department of Mathematics, Princeton University.